Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

TL;DR

This paper compares stochastic volatility and stochastic interest rate models for pricing and hedging Guaranteed Lifelong Withdrawal Benefits (GLWB), introducing four numerical methods to evaluate fees and sensitivities under various strategies.

Contribution

It introduces four numerical methods for pricing GLWB in stochastic models and analyzes their impact on fee calculation and hedging strategies.

Findings

Fees are highly sensitive to economic assumptions.

Optimal withdrawal strategies significantly affect pricing.

Stochastic models provide more accurate valuation than Black-Scholes.

Abstract

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal the Black and Scholes framework seems to be inappropriate for such long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.